RLD Fisher Information Bound for Multiparameter Estimation of … · 2020. 9. 2. · RLD Fisher...

RLD Fisher Information Bound for Multiparameter Estimation of Quantum Channels

Vishal Katariya1 and Mark M. Wilde1, 2

1Hearne Institute for Theoretical Physics, Department of Physics and Astronomy,and Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803, USA

2Stanford Institute for Theoretical Physics, Stanford University, Stanford, California 94305, USA(Dated: September 2, 2020)

One of the fundamental tasks in quantum metrology is to estimate multiple parameters embeddedin a noisy process, i.e., a quantum channel. In this paper, we study fundamental limits to quantumchannel estimation via the concept of amortization and the right logarithmic derivative (RLD) Fisherinformation value. Our key technical result is the proof of a chain-rule inequality for the RLD Fisherinformation value, which implies that amortization, i.e., access to a catalyst state family, does notincrease the RLD Fisher information value of quantum channels. This technical result leads toa fundamental and efficiently computable limitation for multiparameter channel estimation in thesequential setting, in terms of the RLD Fisher information value. As a consequence, we concludethat if the RLD Fisher information value is finite, then Heisenberg scaling is unattainable in themultiparameter setting.

Introduction—Parameter estimation is an information-processing task in which quantum technologies can pro-vide an improvement in performance over the best knownclassical technologies. On one hand, classical parame-ter estimation is limited by the shot-noise limit. On theother hand, quantum parameter estimation can offer asuperclassical “Heisenberg” scaling in estimation error,in principle [1]. Heisenberg scaling refers to the mean-squared error of an estimator scaling as 1/n2, whereasthe shot-noise limit means that the error scales no betterthan 1/n. Here, n refers either to the number of channeluses or to the total probing time allowed in the estima-tion task. One of the fundamental questions in quantumestimation theory is to identify estimation tasks for whichHeisenberg scaling is possible.

One of the most important mathematical tools inestimation theory, both classical and quantum, is theCramer–Rao bound (CRB) [2–4]. It is used to place lowerbounds on the mean squared error of estimators and in-volves an information quantity known as the Fisher infor-mation. The latter captures the amount of informationcarried by a distribution or a quantum channel regardingthe unknown parameter(s). There are an infinite numberof quantum generalizations of the classical Fisher infor-mation, due to the noncommutativity of quantum states.

In this paper, we focus on one quantum generaliza-tion of the Fisher information, known as the right loga-rithmic derivative (RLD) Fisher information [5], and westudy its properties and application in quantum channelestimation. In particular, we prove that the RLD Fisherinformation of quantum channels is a single-letter and ef-ficiently computable multiparameter Cramer–Rao boundfor all quantum channels. Our bound applies in the mostgeneral setting of sequential channel estimation, depictedin Figure 1. “Single-letter” here is a term from informa-tion theory, meaning that the RLD Fisher informationis evaluated with respect to a single copy of the chan-nel only (see (26) for a precise statement of our result).We approach this problem by introducing the amortizedRLD Fisher information value, which quantifies the net

increase in RLD Fisher information that one obtains bysampling from a quantum channel. We prove a chain-rule property for the RLD Fisher information value of aquantum channel, and we then use this to conclude oursingle-letter multiparameter Cramer–Rao bound. Thisbound implies that Heisenberg scaling is unattainable forchannel estimation in the multiparameter setting, when-ever the RLD Fisher information value is finite. Finally,we apply our bound to a concrete example of physicalinterest: estimating the parameters of a generalized am-plitude damping channel.Parameter Estimation—The goal of classical parame-

ter estimation is to obtain an estimate θ of an unknownreal parameter θ embedded in a probability distributionpθ(x), corresponding to a random variable X, where wehave suppressed the dependence of X on θ in our nota-tion. The goal is to guess the value of θ from a realiza-

tion x of the random variable X, and the estimate θ(X) isitself a random variable, being a function of X. It is com-

mon to assume unbiased estimators, i.e., E[θ(X)] = θ.The unbiasedness condition means that the estimator isaccurate and has no systematic error. Hence we focus onstudying the precision of unbiased estimators.

A natural metric to benchmark the performance of an

estimator is the mean-squared error E[(θ(X) − θ)2], ab-breviated as MSE. For an unbiased estimator, the MSE

is equal to the variance Var(θ(X)). The Cramer–Raobound (CRB) places a lower bound on the variance of anunbiased estimator [2–4]:

Var(θ(X)) ≥ 1

IF (θ; {pθ}θ), (1)

where IF (θ; {pθ}θ) is the classical Fisher information, de-fined as

IF (θ; {pθ}θ) := E[(∂θ ln pθ(X))2]. (2)

If one has n independent samples xn described by therandom sequence Xn ≡ X1, . . . , Xn, then the corre-

arX

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008.

1117

8v2

[qu

ant-

ph]

1 S

ep 2

020

2

ρRA1

N θA1B1

S1

N θA2 B2 A3

R1 R2 R3 Rn

S2 · · ·An Bn

N θ

Λθθ

FIG. 1. Processing n uses of channel N θ in a sequential or adaptive manner is the most general approach to channel parameterestimation. The n uses of the channel are interleaved with n quantum channels S1 through Sn−1, which can also share memorysystems with each other. The final measurement’s outcome is then used to obtain an estimate of the unknown parametervector θ.

sponding CRB is

Var(θ(Xn)) ≥ 1

nIF (θ; {pθ}θ). (3)

Estimation theory has been generalized to quantumsystems, where quantum probes and quantum measure-ments are allowed (see, e.g., [6, 7] for recent reviews).We then have an infinite number of logarithmic deriva-tive operators that each reduce to the logarithmic deriva-tive ∂θ ln pθ(x) in the classical case in (2). For single-parameter estimation, one such example is the right loga-rithmic derivative (RLD) operator, defined implicitly viathe differential equation ∂θρθ = ρθRθ, where {ρθ}θ is adifferentiable family of quantum states. The RLD Fisherinformation will be formally defined in the next section.

Consider the task of estimating a real parameter θ en-coded in a quantum channel N θ

A→B (see [8–23] for ex-tensive literature on this problem). The most generalsetting for this problem, given n uses of the channel, con-sists of performing the estimation via a sequential strat-egy [1, 17, 23–26] as depicted in Figure 1. Assuming anunbiased estimator, the following quantum Cramer–Raobound (QCRB) holds in this general setting of channelestimation [27]:

Var(θ) ≥ 1

nIF (θ; {N θA→B}θ)

, (4)

where IF (θ; {N θA→B}θ) is the RLD Fisher information for

the parameter θ encoded in the channel family {N θA→B}θ

[14], and we define it formally later in (9). The resultin (4) is presented in our companion paper [27] and isa special case of the more general result reported herein (26). A restricted subset of the most general sequen-tial strategies consist of parallel strategies, where the nuses of the channel are made simultaneously using an en-tangled probe state, and the bound in (4) was previouslyestablished in [14] for this special case.

The task of simultaneously estimating multiple param-eters is a much more involved task than estimating a sin-gle parameter; however, Cramer–Rao bounds can still beconstructed. This problem, too, has an extensive liter-ature and a number of important recent results [5, 28–43]. See [44, 45] for recent reviews on multiparameterestimation. In the quantum case, an additional com-plication is that the optimal measurements for each pa-rameter may not be compatible. Consider that D pa-rameters need to be estimated and are encoded in a

vector θ := [θ1 θ2 · · · θD]T . For a differentiable fam-ily {ρθ}θ of quantum states, one defines D RLD oper-ators using the differential equations ∂θjρθ = ρθRθj forj ∈ {1, 2, . . . D}. The RLD Fisher information, insteadof being a scalar, takes the form of a D×D matrix with

elements IF (θ; {ρθ}θ)j,k := Tr[R†θjρθRθk ]. This leads to

the following matrix Cramer–Rao bound [5]:

Cov(θ) ≥ IF (θ; {ρθ}θ)−1. (5)

Cov(θ) is a covariance matrix with matrix elements de-fined as

[Cov(θ)]j,k =∑l

Tr[Mlρθ](θj(l)− θj)(θk(l)− θk), (6)

where Ml ≥ 0 are measurement operators satisfying∑lMl = I, and θ(l) := [θ1(l) θ2(l) · · · θD(l)]T is a func-

tion that maps the measurement result l to an estimateof the parameters θ.Right Logarithmic Derivative—The right logarithmic

derivative (RLD) operator leads to one quantum gener-alization of the classical logarithmic derivative [5]. Webegin by defining it for the case of single parameter esti-mation. Consider a single unknown parameter θ embed-ded in a quantum state ρθ. The RLD Fisher informationof the family {ρθ}θ of states is defined as

IF (θ; {ρθ}θ) = Tr[(∂θρθ)2ρ−1θ ], (7)

if supp(∂θρθ) ⊆ supp(ρθ), and it is set to +∞ otherwise.The inverse ρ−1

θ is taken on the support of ρθ. The RLDFisher information is the largest noncommutative Fisherinformation [46]. In some sense, it can also be consideredas the most classical of them.

Alternatively, if the support condition supp(∂θρθ) ⊆supp(ρθ) is satisfied, then the RLD Fisher informationcan also be defined using the RLD operator Rθ as follows:

IF (θ; {ρθ}θ) = Tr[ρθR2θ]. Recall that Rθ is implicitly

defined by the differential equation ∂θρθ = ρθRθ.If, instead of a state family {ρθ}θ, we have a differ-

entiable channel family {N θA→B}θ, then the RLD Fisher

information of this channel family is defined as

IF (θ; {N θA→B}θ) := sup

ρRA

IF (θ; {N θA→B(ρRA)}θ) (8)

=∥∥∥TrB [(∂θΓ

N θRB)(ΓN

θ

RB)−1(∂θΓN θRB)]

∥∥∥∞,

(9)

3

if supp(∂θΓN θRB) ⊆ supp(ΓN

θ

RB), and it is equal to +∞otherwise, where

ΓNθ

RB :=∑i,j

|i〉〈j|R ⊗N θA→B(|i〉〈j|A) (10)

is the Choi operator of the channel N θA→B . The opti-

mization in (8) is with respect to all bipartite input statesρRA that have no dependence on the parameter θ, andthe equality in (9) was established in [14]. Note thatthe optimal value in (8) is achieved by a pure bipartitestate with system R isomorphic to system A. The RLD

Fisher information of a channel family IF (θ; {N θA→B}θ)

is efficiently computable via a semidefinite program [27].The RLD Fisher information of isometric or unitary

channels is an uninteresting information measure. Thisis because a differentiable family {Uθ}θ of isometric orunitary channels induces a differentiable family of purestates. The RLD Fisher information in this case is eitherinfinite or zero, neither of which lead to a useful CRB.

In multiparameter estimation, we have a differentiablefamily {ρθ}θ of quantum states, and the RLD Fisherinformation matrix is given by

IF (θ; {ρθ}θ) :=

D∑j,k=1

Tr[(∂θjρθ)ρ−1θ (∂θkρθ)]|j〉〈k| (11)

if (∂θjρθ)(∂θkρθ)Π⊥ρθ = 0 for all j, k ∈ {1, . . . , D}, where

Π⊥ρθ is the projection onto the kernel of ρθ, and otherwise,

it is undefined (i.e., has some entries equal to +∞). Weprovide more details of the multiparameter RLD Fisherinformation matrix of state families in Appendix A.

As stated in the previous section, the RLD Fisherinformation matrix is featured in the matrix quantumCramer–Rao inequality in (5). To obtain scalar Cramer–Rao bounds from this matrix inequality, we define aD × D positive semidefinite, unit trace weight matrixW (also known as the risk matrix). Then we obtain theRLD Fisher information value, a scalar defined as

IF (θ,W ; {ρθ}θ) := Tr[WIF (θ; {ρθ}θ)] (12)

if∑Dj,k=1〈k|W |j〉(∂θkρθ)(∂θjρθ)Π⊥ρθ = 0 and +∞ other-

wise. By a generalization of [27, Proposition 23], thisquantity is efficiently computable by means of a semi-definite program. We provide the explicit form of thesemi-definite program in Appendix B. Using a weightmatrix W satisfying Tr[W ] = 1, we obtain the follow-ing scalar Cramer–Rao bound:

Tr[WCov(θ)] ≥ 1

nIF (θ,W ; {ρθ}θ), (13)

where n is the number of copies of ρθ available. Weprovide further details and a proof of this bound in Ap-pendix C.

For a differentiable channel family {N θA→B}θ and a

weight matrix W , the RLD Fisher information value isas follows:

IF (θ,W ; {N θA→B}θ) =∥∥∥∥∥∥

D∑j,k=1

〈k|W |j〉TrB [(∂θjΓNθ

RB)(ΓNθ

RB)−1(∂θkΓNθ

RB)]

∥∥∥∥∥∥∞(14)

if

D∑j,k=1

〈k|W |j〉(∂θkΓNθ

RB)(∂θjΓNθ

RB)Π⊥ΓNθ = 0 (15)

and it is equal to +∞ otherwise, where ΓNθ

RB is the Choioperator of the channel N θ

A→B . The quantity in (14) isalso efficiently computable by means of a semi-definiteprogram, as we show in Appendix B.Amortized Fisher Information—Amortized channel di-

vergences were defined in [47] to provide a mathematicalframework for studying the power of sequential strate-gies over parallel ones in quantum channel discrimina-tion. With the view of performing a similar comparisonfor quantum channel estimation, we define the amortizedFisher information for quantum channel families [27]. Forthe channel family {N θ

A→B}θ, the amortized RLD Fisherinformation is defined as

IAF (θ; {N θA→B}θ) :=

sup{ρθRA}θ

[IF (θ; {N θ

A→B(ρθRA)}θ)− IF (θ; {ρθRA}θ)]. (16)

For the case of multiparameter estimation, we define theamortized RLD Fisher information value as follows:

IAF (θ,W ; {N θA→B}θ) :=

sup{ρθRA}θ

IF (θ,W ; {N θA→B(ρθRA)}θ)− IF (θ,W ; {ρθRA}θ).

(17)

The framework of amortized Fisher information can beapplied more generally beyond RLD Fisher information,as discussed in our companion paper [27], and it is help-ful in clarifying the power of sequential strategies whenestimating a parameter encoded in a quantum channel.In the sequential strategies described in Figure 1, onecan qualitatively say that the goal is to “accumulate” asmuch information about θ into the state being carriedforward from one channel use to another. The amor-tized Fisher information captures the marginal increasein Fisher information per channel use in such a scenario.

A basic inequality obeyed by the amortized RLDFisher information, as a direct consequence of definitionsin (8) and (16), is that

IAF (θ; {N θA→B}θ) ≥ IF (θ; {N θ

A→B}θ). (18)

This can be easily understood by considering the right-hand side to arise from restricting to input states with

4

no parameter dependence. It also holds for the multipa-rameter case:

IAF (θ,W ; {N θA→B}θ) ≥ IF (θ,W ; {N θ

A→B}θ). (19)

For the RLD Fisher information of quantum channels,we actually have something much stronger, what is calledan “amortization collapse,” i.e.,

IAF (θ; {N θA→B}θ) = IF (θ; {N θ

A→B}θ), (20)

IAF (θ,W ; {N θA→B}θ) = IF (θ,W ; {N θ

A→B}θ). (21)

The meaning of an amortization collapse is that the RLDFisher information cannot be increased by using a cat-alyst. That is, a catalyst state family {ρθRA}θ at thechannel input does not increase the RLD Fisher informa-

tion IF (θ,W ; {N θA→B}θ) by any more than its own RLD

Fisher information IF (θ,W ; {ρθRA}θ). Continuing theearlier discussion of amortized Fisher information cap-turing marginal increment in resource, the amortizationcollapse in (20)–(21) means that the RLD Fisher infor-mation gained during each of the n sequential uses of thechannel N θ

A→B is the same. To establish the amortiza-tion collapse for the RLD Fisher information, we prove achain-rule inequality, as discussed below.

Chain Rule—Let {N θA→B}θ be a differentiable family

of quantum channels, and let {ρθRA}θ be a differentiablefamily of quantum states on systems RA, where the sys-tem R can be of arbitrary size. Then the following chainrule holds [27]

IF (θ; {N θA→B(ρθRA)}θ) ≤ IF (θ; {N θ

A→B}θ)

+ IF (θ; {ρθRA}θ), (22)

analogous to one that holds for a different distinguisha-bility measure [48]. This can be generalized to the case ofestimating multiple parameters. With W a weight ma-trix as defined above, let {N θ

A→B}θ be a differentiablefamily of quantum channels, and let {ρθRA}θ be a differ-entiable family of quantum states. Then the followingchain-rule inequality holds

IF (θ,W ; {N θA→B(ρθRA)}θ) ≤ IF (θ,W ; {N θ

A→B}θ)

+ IF (θ,W ; {ρθRA}θ), (23)

as proved in Appendix D.We now show how the chain rule results in an amorti-

zation collapse for the mathematically simpler case of asingle parameter. The chain rule in (22) can be rewrittenas

IF (θ; {N θA→B(ρθRA)}θ)− IF (θ; {ρθRA}θ)

≤ IF (θ; {N θA→B}θ). (24)

Taking the supremum over input state families on the

left-hand side gives IAF (θ; {N θA→B}θ) ≤ IF (θ; {N θ

A→B}θ).

When the above is combined with (18), we obtain thedesired amortization collapse in (20).

The same reasoning can be repeated for the case of es-timating multiple parameters. That is, the chain rule forthe multiparameter RLD Fisher information (23) leads tothe amortization collapse in (21). The amortization col-lapse for the RLD Fisher information implies that the n-sequential-use RLD Fisher information is simply n timesthe single-use RLD Fisher information.Single-letter quantum Cramer–Rao Bounds—The fol-

lowing single-letter Cramer–Rao bound for estimatingparameter θ embedded in {N θ

A→B}θ holds both in theparallel setting [14] and the more general sequential set-ting [27] of channel estimation. In the sequential setting,it arises from the combination of the chain rule, the en-suing amortization collapse for the RLD Fisher informa-tion, and Theorem 18 of [27]:

Var(θ) ≥ 1

nIF (θ; {N θA→B}θ)

. (25)

For the multiparameter case, with differentiable chan-nel family {N θ

A→B}θ and a weight matrix W , we usethe chain rule, ensuing amortization collapse, as well asTheorem 6 (stated and proved in Appendix E), to ob-tain the following efficiently computable, multiparameterCramer–Rao bound:

Tr[WCov(θ)] ≥ 1

nIF (θ,W ; {N θA→B}θ)

. (26)

The bounds in (25) and (26) have an importantimplication: if the RLD Fisher information is fi-nite, then Heisenberg scaling is impossible. That is,

if supp(∂θΓN θRB) ⊆ supp(ΓN

θ

RB) holds for the single-parameter case or if (15) holds for the multiparametercase, then every estimator is limited by the shot-noiselimit. A simple corollary of this is that for every full-rankquantum channel family, Heisenberg scaling is unattain-able. This complements results on this problem fromother works [12–14, 19, 20, 23].Evaluating bounds for Generalized Amplitude Damp-

ing Channel—Here, we apply the Cramer–Rao boundsderived in (25) and (26) to the task of estimating theparameters of a generalized amplitude damping channel[49], a qubit-to-qubit channel that is used to model noisein, e.g., superconducting quantum circuits. A general-ized amplitude damping channel (GADC) is defined interms of its loss parameter γ ∈ (0, 1) and noise parame-ter N ∈ (0, 1).

The Choi operator of a GADC Aγ,N with loss param-eter γ and noise parameter N is given by

ΓAγ,NRB :=

1− γN 0 0√

1− γ0 γN 0 00 0 γ (1−N) 0√

1− γ 0 0 1− γ (1−N)

.(27)

5

0 0.2 0.4 0.6 0.8 1-4

-3

-2

-1

0

1

2

3Lo

garit

hm o

f low

er b

ound

val

ueRLD lower boundSLD lower bound

FIG. 2. Logarithm of RLD and SLD Fisher informationvalues versus loss γ for noise N = 0.2. Both lines indi-cate lower bounds on the quantity Tr[WCov({γ,N})] whereW = 1

4

(1 11 3

)(as discussed in the main text, the SLD Fisher

information value is a lower bound up to a constant prefac-tor). For the SLD quantity, we optimize over input states ofthe form

√p|00〉 +

√1− p|11〉.

Consider that we wish to estimate both parameters γand N simultaneously. Let us choose W = 1

4

(1 11 3

). To

compute the RLD bound in (26), we calculate

IF ({γ,N},W ; {Aγ,N}γ,N ) =

1

4

∥∥∥TrB [(∂γΓA)(ΓA)−1(∂γΓA) + (∂γΓA)(ΓA)−1(∂NΓA)

+(∂NΓA)(ΓA)−1(∂γΓA)+3(∂NΓA)(ΓA)−1(∂NΓA)]∥∥∥∞.

(28)

where A is used as shorthand for Aγ,N . As a conse-quence of (26), the inverse of (28) is a lower bound onTr[WCov({γ,N})].

We compare the RLD Fisher information bound to thegeneralized Helstrom Cramer–Rao bound [40], which inthis case reduces to the SLD Fisher information bound.This lower bound can be achieved asymptotically up toa constant prefactor [38, 50–53].

To calculate the SLD Fisher information, we chooseinput probe state |ψ(p)〉 :=

√p|00〉 +

√1− p|11〉, with

W chosen as above. It suffices to optimize the single-copy SLD Fisher information over such states due to theσZ covariance of the channel Aγ,N . We optimize overparameter p to obtain the SLD Fisher information bound(to be clear, we minimize

Tr[W [IF ({γ,N}; {Aγ,N (ψ(p)))}{γ,N}]−1] (29)

with respect to p). We compare the two bounds in Fig-ure 2, where we keep N fixed and vary γ from 0 to 1. Fig-ure 2 demonstrates that the RLD lower bound is withinone to two orders of magnitude of the SLD Fisher infor-mation. Full details of this calculation can be found inAppendix F.

Conclusion—In this paper, we provided a single-letter,efficiently computable Cramer–Rao bound for the taskof multiparameter estimation. We did so by introducingthe amortized RLD Fisher information and proving anamortization collapse for it. The single-letter Cramer–Rao bound also leads to a simple yet rigorous no-go con-dition for Heisenberg scaling in quantum multiparameterestimation. Finally, we evaluated our bound when esti-mating the two parameters of a generalized amplitudedamping channel.

Lately, multiparameter estimation has been a fruitfularea of study in quantum information theory. We believethat our results both complement and extend the currentbody of work, and so they should have metrological ap-plications theoretically and experimentally.

ACKNOWLEDGMENTS

This paper is dedicated to the memory of Jonathan P.Dowling. Jon’s impact on quantum metrology was far-reaching, and he played a role in inspiring this project.We acknowledge Pieter Kok, Jasminder Sidhu, and SisiZhou for helpful discussions. We thank Masahito Hayashifor feedback on our paper. VK acknowledges sup-port from the LSU Economic Development Assistantship.VK and MMW acknowledge support from the US Na-tional Science Foundation via grant number 1907615.MMW acknowledges support from Stanford QFARM andAFOSR (FA9550-19-1-0369).

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8

Supplementary Material for “RLD Fisher Information Bound forMultiparameter Estimation of Quantum Channels”

by Vishal Katariya and Mark M. Wilde

Appendix A: RLD Fisher Information

Let {ρθ}θ be a differentiable family of d-dimensional quantum states, and suppose that the pa-rameter vector θ contains D elements. In the case that the following finiteness conditions hold

(∂θjρθRA)(∂θkρ

θRA)Π⊥ρθ = 0 ∀j, k ∈ {1, . . . , D} (A1)

the matrix elements of the multiparameter RLD Fisher information matrix are defined as follows:

[IF (θ; {ρθ}θ)]j,k := Tr[(∂θjρθ)ρ−1θ (∂θkρθ)], (A2)

Then the RLD Fisher information matrix is defined as

IF (θ; {ρθ}θ) :=D∑

j,k=1

Tr[(∂θjρθ)ρ−1θ (∂θkρθ)]|j〉〈k| (A3)

= Tr2

[D∑

j,k=1

|j〉〈k| ⊗ (∂θjρθ)ρ−1θ (∂θkρθ)

]. (A4)

If the finiteness conditions in (A1) do not hold, then the RLD matrix is undefined (some or all of itsentries are equal to +∞). The analysis for this statement is similar to that given in our companionpaper [27, Proposition 4].

Suppose that the finiteness conditions in (A1) hold. Let Rθj denote the RLD operator satisfyingthe following differential equation:

∂θjρθ = ρθRθj . (A5)

We find upon substitution that

[IF (θ; {ρθ}θ)]j,k = Tr[(∂θjρθ)ρ−1θ (∂θkρθ)] (A6)

= Tr[(∂θjρθ)†ρ−1θ (∂θkρθ)] (A7)

= Tr[(ρθRθj)†ρ−1

θ (ρθRθk)] (A8)

= Tr[R†θjρθRθk ]. (A9)

If we define a D × D positive semi-definite weight matrix W , then the RLD Fisher informationvalue is given by

IF (θ,W ; {ρθ}θ) := Tr[WIF (θ; {ρθ}θ)] (A10)

= Tr

[(W ⊗ Id)

(D∑

j,k=1

|j〉〈k| ⊗ (∂θjρθ)ρ−1θ (∂θkρθ)

)](A11)

=D∑

j,k=1

〈k|W |j〉Tr[(∂θjρθ)ρ−1

θ (∂θkρθ)]. (A12)

9

The same quantity for a differentiable family {N θA→B}θ of quantum channels, is defined as

IF (θ,W ; {N θA→B}θ) := sup

ρRA

IF (θ,W ; {N θA→B(ρRA)}θ), (A13)

where the optimization is with respect to every bipartite state ρRA with system R arbitrarily large.However, note that, by a standard argument, it suffices to optimize over pure states ψRA with systemR isomorphic to the channel input system A.

The finiteness conditions for the state and channel quantities, respectively, involving the weightmatrix W are [

D∑j,k=1

〈k|W |j〉(∂θkρθ)(∂θjρθ)

]Π⊥ρθ = 0, (A14)[

D∑j,k=1

〈k|W |j〉(∂θkΓNθ

RB)(∂θjΓNθ

RB)

]Π⊥

ΓNθ = 0. (A15)

The argument for the first finiteness condition is similar to that given for (A1), while the argumentfor the second relies on a generalization of the argument in [27, Proposition 13].

Proposition 1 Let {N θA→B}θ be a differentiable family of quantum channels, and let W be a D×D

weight matrix. Suppose that (A15) holds. Then the RLD Fisher information value of quantumchannels has the following explicit form:

IF (θ,W ; {N θA→B}θ) =

∥∥∥∥∥D∑

j,k=1

〈k|W |j〉TrB[(∂θjΓNθ

RB)(ΓNθ

RB)−1(∂θkΓNθ

RB)]

∥∥∥∥∥∞

, (A16)

where ΓNθ

RB is the Choi operator of the channel N θA→B.

Proof. Recall that every pure state ψRA can be written as

ψRA = ZRΓRAZ†R, (A17)

where ZR is a square operator satisfying Tr[Z†RZR] = 1. This implies that

N θA→B(ψRA) = N θ

A→B(ZRΓRAZ†R) (A18)

= ZRN θA→B(ΓRA)Z†R (A19)

= ZRΓNθ

RBZ†R. (A20)

It suffices to optimize over pure states ψRA such that ψA > 0 because these states are dense in theset of all pure bipartite states. Then consider that

supψRA

IF (θ,W ; {N θA→B(ψRA)}θ)

= supψRA

D∑j,k=1

〈k|W |j〉Tr[(∂θjN θ

A→B(ψRA))(N θA→B(ψRA))−1(∂θkN θ

A→B(ψRA))]

(A21)

= supZR:Tr[Z†RZR]=1

D∑j,k=1

〈k|W |j〉Tr[(∂θjZRΓN

θ

RBZ†R)(ZRΓN

θ

RBZ†R)−1(∂θkZRΓN

θ

RBZ†R)]

(A22)

10


D∑j,k=1

〈k|W |j〉Tr[ZR(∂θjΓ

Nθ

RB)Z†RZ−†R (ΓN

θ

RB)−1Z−1R ZR(∂θkΓ

Nθ

RB)Z†R

](A23)


D∑j,k=1

〈k|W |j〉Tr[Z†RZR(∂θjΓ

Nθ

RB)(ΓNθ

RB)−1(∂θkΓNθ

RB)]

(A24)


Tr

[Z†RZR

D∑j,k=1

〈k|W |j〉TrB

[(∂θjΓ

Nθ

RB)(ΓNθ

RB)−1(∂θkΓNθ

RB)]]

(A25)

=

∥∥∥∥∥D∑

j,k=1

〈k|W |j〉TrB

[(∂θjΓ

Nθ

RB)(ΓNθ

RB)−1(∂θkΓNθ

RB)]∥∥∥∥∥∞

(A26)

The last equality is a consequence of the characterization of the infinity norm of a positive semi-definite operator Y as ‖Y ‖∞ = supρ>0,Tr[ρ]=1 Tr[Y ρ].

Appendix B: Semi-definite Programs for the RLD Fisher Information Value of States and Channels

Proposition 2 Let {ρθA}θ be a differentiable family of quantum states, and let WF be a D×D weightmatrix. Suppose that (A14) holds. Then the RLD Fisher information value of quantum states canbe calculated via the following semi-definite program:

IF (θ,W ; {ρθA}) = inf{

Tr[(WF ⊗ IA)MFA] : MFA ≥ 0,[MFA

∑Dj=1 |j〉F ⊗ (∂θjρ

θA)∑D

j=1〈j|F ⊗ (∂θjρθA) ρθA

]≥ 0}. (B1)

The dual program is given by

supPFA,QFA,RA

2

(D∑j=1

Re[Tr[Q(|j〉F ⊗ (∂θjρθA))]]

)− Tr[RρθA], (B2)

subject to

PFA ≤ (WF ⊗ IA),

[PFA Q†FAQFA RA

]≥ 0, (B3)

where PFA and RA are Hermitian, and QFA is a linear operator.

Proof. We begin with the formula

I(θ,W ; {ρθ}θ) := Tr

[(WF ⊗ IA)

(D∑

j,k=1

|j〉〈k|F ⊗ (∂θjρθA)(ρθA)−1(∂θkρ

θA)

)](B4)

=D∑

j,k=1

〈k|W |j〉Tr[(∂θjρ

θA)(ρθA)−1(∂θkρ

θA)]. (B5)

The above can be written as

Tr[(WF ⊗ IA)(X†Y −1X)

](B6)

11

where X =∑D

j=1〈j|F ⊗ (∂θjρθA) and Y = ρθA.

We next use the Schur complement lemma,[M X†

X Y

]≥ 0 ⇐⇒ Y ≥ 0, M ≥ X†Y −1X (B7)

which lets us write

X†Y −1X = min

{M :

[M X†

X Y

]≥ 0

}, (B8)

where the ordering is understood in the Lowner sense.Combining the above, we obtain the desired primal form in (B1).To obtain the dual program in (B2), we apply [27, Lemma 53].

Proposition 3 Let {N θA→B}θ be a differentiable family of quantum channels, and let WF be a D×D

weight matrix. Suppose that (A15) holds. Then the RLD Fisher information value of quantumchannels can be calculated via the following semi-definite program:

IF (θ,W ; {N θA→B}θ) = inf λ ∈ R+, (B9)

subject to

λIR ≥ TrFB[(WF ⊗ IRB)MFRB],

[MFRB

∑j |j〉F ⊗ (∂θjΓ

Nθ

RB)∑j〈j|F ⊗ (∂θjΓ

Nθ

RB) ΓNθ

RB

]≥ 0. (B10)

The dual program is given by

supρR≥0,PFRB ,ZFRB ,QRB

2

(D∑j=1

Re[Tr[ZFRB(|j〉F ⊗ (∂θjΓNθ

RB))]]

)− Tr[QRBΓN

θ

RB], (B11)

subject to

Tr[ρR] ≤ 1,

[PFRB Z†FRBZFRB QRB

]≥ 0, PFRB ≤ ρR ⊗ IFB. (B12)

Proof. The form of the primal program relies on the combination of a few facts. First, we use thefollowing characterization of the infinity norm of a positive semi-definite operator A:

‖A‖∞ = inf {λ ≥ 0 : A ≤ λI} . (B13)

Next we observe that

D∑j,k=1


RB)(ΓNθ

RB)−1(∂θkΓNθ

RB)] (B14)

can be written as

TrFB[(WF ⊗ IRB)(X†Y −1X)

](B15)

where X =∑D

j=1〈j|F ⊗ (∂θjΓNθ

RB) and Y = ΓNθ

RB.

12

We next use the Schur complement lemma,[M X†

X Y

]≥ 0 ⇐⇒ Y ≥ 0, M ≥ X†Y −1X (B16)

which lets us write

X†Y −1X = min

{M :

[M X†

X Y

]≥ 0

}(B17)

Combining the above, we obtain the desired primal form in (B9).To arrive at the dual program, we use the standard forms of primal and dual semi-definite programs

for Hermitian operators A and B and a Hermiticity-preserving map Φ [54]:

supX≥0{Tr[AX] : Φ(X) ≤ B} , inf

Y≥0

{Tr[BY ] : Φ†(Y ) ≥ A

}. (B18)

From (B9)–(B10), we identify

B =

[1 00 0

], Y =

[λ 00 MFRB

], Φ†(Y ) =

λIR − TrFB[MFRB] 0 00 MFRB 00 0 0

, (B19)

A =

0 0 0

0 0 −∑D

j=1 |j〉F ⊗ (∂θjΓNθ

RB)

0 −∑D


RB) −ΓNθ

RB

. (B20)

Upon setting

X =

ρR 0 0

0 PFRB Z†FRB0 ZFRB QRB

, (B21)

we find that

Tr[XΦ†(Y )] = Tr

ρR 0 0


λIR − TrFB[MFRB] 0 00 MFRB 00 0 0

(B22)

= Tr[ρR(λIR − TrFB[MFRB])] + Tr[PFRBMFRB] (B23)

= λTr[ρR] + Tr[(PFRB − ρR ⊗ IFB)MFRB] (B24)

= Tr

[[λ 00 MFRB

] [Tr[ρR] 0

0 PFRB − ρR ⊗ IFB

]], (B25)

which implies that

Φ(X) =

[Tr[ρR] 0


]. (B26)

Then plugging into the left-hand side of (B18), we find that the dual is given by

supρR,PFRB ,ZFRB ,QRB

Tr

0 0 0

0 0 −∑D

j=1 |j〉F ⊗ (∂θjΓNθ

RB)

0 −∑D


RB) −ΓNθ

RB

ρR 0 0


, (B27)

13

subject to ρR 0 0


≥ 0,

[Tr[ρR] 0


]≤[1 00 0

]. (B28)

We can swap ZFRB → −ZFRB with no change to the optimal value. This leads to the followingsimplified form of the dual program:

supρR≥0,PFRB ,ZFRB ,QRB

2D∑j=1

Re[Tr[ZFRB(|j〉F ⊗ ∂θjΓNθ

RB)]]− Tr[QRBΓNθ

RB], (B29)

subject to

Tr[ρR] ≤ 1,

[PFRB −Z†FRB−ZFRB QRB

]≥ 0, PFRB ≤ ρR ⊗ IFB. (B30)

Then we note that [PFRB −Z†FRB−ZFRB QRB

]≥ 0 ⇐⇒

[PFRB Z†FRBZFRB QRB

]≥ 0 (B31)

This concludes the proof.

Appendix C: Multiparameter Cramer–Rao bound

We start with the matrix inequality [6, 55, 56]

Cov(θ) ≥ IF (θ; {ρθ}θ)−1. (C1)

Let us take W as a positive semi-definite weight matrix, and then define the normalized operator W ′

as

W ′ :=W

Tr[W ]. (C2)

The matrix inequality in (C1) implies the following:

Tr[W ] Tr[W ′Cov(θ)] ≥ Tr[W ] Tr[W ′IF (θ; {ρθ}θ)−1] (C3)

= Tr[W ] Tr[W ′1/2IF (θ; {ρθ}θ)−1W ′1/2] (C4)

= Tr[W ]∑k

〈k|W ′1/2IF (θ; {ρθ}θ)−1W ′1/2|k〉 (C5)

≥ Tr[W ]

[∑k

〈k|W ′1/2IF (θ; {ρθ}θ)W ′1/2|k〉

]−1

(C6)

=Tr[W ]

Tr[W ′1/2IF (θ; {ρθ}θ)W ′1/2](C7)

=Tr[W ]

Tr[W ′IF (θ; {ρθ}θ)](C8)

14

=Tr[W ]2

Tr[WIF (θ; {ρθ}θ)]. (C9)

The first inequality is a consequence of (C1). The first equality follows from cyclicity of trace.The second inequality uses the operator Jensen inequality [57] for the operator convex functionf(x) = x−1. The next equality comes again from cyclicity of trace, and the last equality comes fromthe definition of W ′.

The reasoning above leads us to

Tr[WCov(θ)] ≥ (Tr[W ])2

Tr[WIF (θ; {ρθ}θ)](C10)

for every positive semi-definite matrix W , which implies that

Tr[W ′Cov(θ)] ≥ 1

Tr[W ′IF (θ; {ρθ}θ)](C11)

for every positive semi-definite matrix W ′ such that Tr[W ′] = 1.

Appendix D: Chain Rule for RLD Fisher Information

In this appendix, we provide a proof of the chain-rule property for the RLD Fisher informationvalue, as stated in the main text (Eq. (23)). First, we establish the chain rule in the form of anoperator inequality:

Proposition 4 Let {N θA→B}θ be a differentiable family of quantum channels, and let {ρθRA}θ be a

differentiable family of quantum states. Then the following chain-rule operator inequality holds

IF (θ; {N θA→B(ρθRA)}θ) ≤

D∑j,k=1

|j〉〈k|Tr[(ρθS)T TrB[(∂θjΓNθ

SB )(ΓNθ

SB )−1(∂θkΓNθ

SB )] + IF (θ; {ρθRA}θ), (D1)

where ρθS is equal to the reduced state of ρθRA on system A and system S is isomorphic to system A.

Proof. In the proof, we make use of the following identity:

N θA→B(ρθRA) = 〈Γ|ASρθRA ⊗ ΓN

θ

SB |Γ〉AS, (D2)

where

|Γ〉AS :=∑i

|i〉A|i〉S. (D3)

Consider that

IF (θ; {N θA→B(ρθRA)}θ) = Tr2

[D∑

j,k=1

|j〉〈k| ⊗ (∂θjN θA→B(ρθRA))(N θ

A→B(ρθRA))−1(∂θkN θA→B(ρθRA))

].

(D4)So we now focus on the operator inside the trace:

D∑j,k=1



16

= ((∂θjρθRA)⊗ ΓN

θ


SB ))((ρθRA)−1 ⊗ (ΓNθ

SB )−1) (D16)

= (∂θjρθRA)(ρθRA)−1 ⊗ ΓN

θ

SB (ΓNθ

SB )−1 + ρθRA(ρθRA)−1 ⊗ (∂θjΓNθ

SB )(ΓNθ

SB )−1 (D17)

= (∂θjρθRA)(ρθRA)−1 ⊗ Π

ΓNθ + Πρθ ⊗ (∂θjΓNθ

SB )(ΓNθ

SB )−1. (D18)

Right multiplying this last line by ∂θk(ρθRA ⊗ ΓN

θ

SB ) gives

((∂θjρθRA)(ρθRA)−1 ⊗ Π


SB )(ΓNθ

SB )−1)∂θk(ρθRA ⊗ ΓN

θ

SB )

= ((∂θjρθRA)(ρθRA)−1 ⊗ Π


SB )(ΓNθ

SB )−1)((∂θkρθRA)⊗ ΓN

θ

SB + ρθRA ⊗ (∂θkΓNθ

SB )) (D19)

= ((∂θjρθRA)(ρθRA)−1 ⊗ Π

ΓNθ )((∂θkρθRA)⊗ ΓN

θ

SB ) + ((∂θjρθRA)(ρθRA)−1 ⊗ Π

ΓNθ )(ρθRA ⊗ (∂θkΓNθ

SB ))

+ (Πρθ ⊗ (∂θjΓNθ

SB )(ΓNθ

SB )−1)((∂θkρθRA)⊗ ΓN

θ

SB ) + (Πρθ ⊗ (∂θjΓNθ

SB )(ΓNθ

SB )−1)(ρθRA ⊗ (∂θkΓNθ

SB ))(D20)

= (∂θjρθRA)(ρθRA)−1(∂θkρ

θRA)⊗ ΓN

θ

SB + (∂θjρθRA)Πρθ ⊗ Π

ΓNθ (∂θkΓNθ

SB )

+ Πρθ(∂θkρθRA)⊗ (∂θjΓ

Nθ

SB )ΠΓNθ + ρθRA ⊗ (∂θjΓ

Nθ

SB )(ΓNθ


SB ). (D21)

Adding in the terms (∂θjρθRA)Π⊥

ρθ= 0, (∂θjΓ

Nθ

SB )Π⊥ΓNθ = 0, Π⊥

ρθ(∂θkρ

θRA) = 0, and Π⊥

ΓNθ (∂θkΓNθ

SB ) = 0,

which follow from the support conditions for finite RLD Fisher information, the last line abovebecomes as follows:

= (∂θjρθRA)(ρθRA)−1(∂θkρ

θRA)⊗ ΓN

θ

SB + (∂θjρθRA)⊗ (∂θkΓ

Nθ

SB )

+ (∂θkρθRA)⊗ (∂θjΓ

Nθ

SB ) + ρθRA ⊗ (∂θjΓNθ

SB )(ΓNθ


SB ). (D22)

So then the relevant matrix is simplified as follows:

D∑j,k=1

|j〉〈k| ⊗ 〈Γ|AS[(∂θjρθRA)(ρθRA)−1(∂θkρ

θRA)⊗ ΓN

θ


Nθ

SB )


Nθ


SB )(ΓNθ


SB )]|Γ〉AS. (D23)

Thus, we have established the following operator inequality:

D∑j,k=1



≤D∑

j,k=1

|j〉〈k| ⊗ 〈Γ|AS[(∂θjρθRA)(ρθRA)−1(∂θkρ

θRA)⊗ ΓN

θ


Nθ

SB )


Nθ


SB )(ΓNθ


SB )]|Γ〉AS. (D24)

We can then take a partial trace over the RB systems and arrive at the following operator inequality:

IF (θ; {N θA→B(ρθRA)}θ)

≤D∑

j,k=1

|j〉〈k| TrRB[〈Γ|AS[(∂θjρθRA)(ρθRA)−1(∂θkρ

θRA)⊗ ΓN

θ


Nθ

SB )


Nθ


SB )(ΓNθ


SB )]|Γ〉AS]. (D25)

17

Now we evaluate each term on the right:

〈Γ|AS TrRB[(∂θjρθRA)(ρθRA)−1(∂θkρ

θRA)⊗ ΓN

θ

SB ]|Γ〉AS= 〈Γ|AS TrR[(∂θjρ

θRA)(ρθRA)−1(∂θkρ

θRA)]⊗ TrB[ΓN

θ

SB ]|Γ〉AS (D26)

= 〈Γ|AS TrR[(∂θjρθRA)(ρθRA)−1(∂θkρ

θRA)]⊗ IS|Γ〉AS (D27)

= Tr[(∂θjρθRA)(ρθRA)−1(∂θkρ

θRA)], (D28)

〈Γ|AS TrRB[(∂θkρθRA)⊗ (∂θjΓ

Nθ

SB )]|Γ〉AS= 〈Γ|AS TrR[∂θkρ

θRA]⊗ TrB[∂θjΓ

Nθ

SB ]|Γ〉AS (D29)

= 〈Γ|AS TrR[∂θkρθRA]⊗ ∂θj TrB[ΓN

θ

SB ]|Γ〉AS (D30)

= 〈Γ|AS TrR[∂θkρθRA]⊗ (∂θjIS)|Γ〉AS (D31)

= 0, (D32)

〈Γ|AS TrRB[(∂θjρθRA)⊗ (∂θkΓ

Nθ

SB )]|Γ〉AS = 0, (D33)

〈Γ|AS TrRB[ρθRA ⊗ (∂θjΓNθ

SB )(ΓNθ


SB )]|Γ〉AS= 〈Γ|ASρθA ⊗ TrB[(∂θjΓ

Nθ

SB )(ΓNθ


SB )]|Γ〉AS (D34)

= Tr[(ρθS)T TrB[(∂θjΓNθ

SB )(ΓNθ


SB )]. (D35)

Substituting back above, we find that

IF (θ; {N θA→B(ρθRA)}θ)

≤D∑

j,k=1

|j〉〈k| Tr[(∂θjρθRA)(ρθRA)−1(∂θkρ

θRA)]

+D∑

j,k=1

|j〉〈k| Tr[(ρθS)T TrB[(∂θjΓNθ

SB )(ΓNθ


SB )] (D36)

= IF (θ; {ρθRA}θ) +D∑

j,k=1

|j〉〈k| Tr[(ρθS)T TrB[(∂θjΓNθ

SB )(ΓNθ


SB )]. (D37)


Now we show how the operator-inequality chain rule leads us to the desired result in (23).

Proposition 5 Let W be a D×D weight matrix, let {N θA→B}θ be a differentiable family of quantum

channels, and let {ρθRA}θ be a differentiable family of quantum states. Then the following chain-ruleinequality holds


A→B}θ) + IF (θ,W ; {ρθRA}θ). (D38)

Proof. We start by restating the operator-inequality chain rule in (D1):

IF (θ; {N θA→B(ρθRA)}θ) ≤

D∑j,k=1


SB )(ΓNθ


SB )] + IF (θ; {ρθRA}θ). (D39)

We sandwich the above with W 1/2 on both sides and take the trace (a positive map overall) topreserve the inequality and obtain

18

Tr[WIF (θ; {N θA→B(ρθRA)}θ)] ≤ Tr

[W

D∑j,k=1


SB )(ΓNθ


SB )]]

]+ Tr[WIF (θ; {ρθRA}θ)]. (D40)

Using the definition of the RLD Fisher information value, the above simplifies to

IF (θ,W ; {N θA→B(ρθRA)}θ) ≤ Tr

[W

D∑j,k=1


SB )(ΓNθ


SB )]]

]+ IF (θ,W ; {ρθRA}θ). (D41)

Now we consider that

Tr

[W

D∑j,k=1


SB )(ΓNθ


SB )]

]

=D∑

j,k=1

〈k|W |j〉Tr[(ρθS)T TrB[(∂θjΓ

Nθ

SB )(ΓNθ


SB )]]

(D42)

= Tr

[(ρθS)T

D∑j,k=1


SB )(ΓNθ


SB )]

](D43)

≤ supρθS

Tr

[(ρθS)T

D∑j,k=1


SB )(ΓNθ


SB )]

](D44)

=

∥∥∥∥∥D∑

j,k=1


SB )(ΓNθ


SB )]

∥∥∥∥∥∞

(D45)

= IF (θ,W ; {N θA→B}θ. (D46)

Using the above and (D41), we conclude that


A→B}θ + IF (θ,W ; {ρθRA}θ). (D47)


Appendix E: Converse for RLD Fisher Information

The following inequality is a multi-parameter generalization of the inequality given in [27, Theo-rem 18].

Theorem 6 Consider a general sequential estimation protocol of the form depicted in Figure 1. Thefollowing inequality holds:

IF (θ,W ; {ωθRnBn}θ) ≤ n · IAF (θ,W ; {N θ

A→B}θ), (E1)

where ωθRnBn

is the final state of the estimation protocol.

19

Proof. Consider that

IF (θ,W ; {ωθRnBn}θ)

= IF (θ,W ; {ωθRnBn}θ)− IF (θ,W ; {ρR1A1}θ) (E2)

= IF (θ,W ; {ωθRnBn}θ)− IF (θ,W ; {ρR1A1}θ) +

n∑i=2

(IF (θ,W ; {ρθRiAi}θ)− IF (θ,W ; {ρθRiAi}θ)

)(E3)

= IF (θ,W ; {ωθRnBn}θ)− IF (θ,W ; {ρR1A1}θ)

+n∑i=2

(IF (θ,W ; {S i−1

Ri−1Bi−1→RiAi(ρθRi−1Bi−1

)}θ)− IF (θ,W ; {ρθRiAi}θ))

(E4)

≤ IF (θ,W ; {ωθRnBn}θ)− IF (θ,W ; {ρR1A1}θ)

+n∑i=2

(IF (θ,W ; {ρθRi−1Bi−1

}θ)− IF (θ,W ; {ρθRiAi}θ))

(E5)

=n∑i=1

(IF (θ,W ; {ρθRiBi}θ)− IF (θ,W ; {ρθRiAi}θ

)(E6)

=n∑i=1

(IF (θ,W ; {N θ

Ai→Bi(ρθRiAi

)}θ)− IF (θ,W ; {ρθRiAi}θ)

(E7)

≤ n · sup{ρθRA}θ

[IF (θ,W ; {N θ

A→B(ρθRA)}θ)− IF (θ,W ; {ρθRA)}θ)]

(E8)

= n · IAF (θ,W ; {N θA→B}θ). (E9)

The first equality follows because the initial state ρR1A1 has no dependence on any of the parametersin θ. The first inequality arises due to the data-processing inequality for the RLD Fisher information.The other steps are straightforward manipulations.

Appendix F: Estimating the parameters of a generalized amplitude damping channel

For completeness, we restate the Choi matrix of a generalized amplitude damping channel (GADC)Aγ,N with parameters γ and N :

ΓAγ,NRB :=

1− γN 0 0

√1− γ

0 γN 0 00 0 γ (1−N) 0√

1− γ 0 0 1− γ (1−N)

. (F1)

This means that

(ΓAγ,NRB

)−1

=

1−γ(1−N)(1−N)Nγ2

0 0 −√

1−γ(1−N)Nγ2

0 1γN

0 0

0 0 1γ(1−N)

0−√

1−γ(1−N)Nγ2

0 0 1−γN(1−N)Nγ2

. (F2)

20

1. RLD Fisher Information Value

We use that

IF (θ,W ; {N θA→B}θ) =

∥∥∥∥∥D∑

j,k=1


RB)(ΓNθ

RB)−1(∂θkΓNθ

RB)]

∥∥∥∥∥∞

. (F3)

This is proved in Proposition 1 in Appendix A.We consider W = 1

4

(1 11 3

)which means that

IF ({γ,N};W ; {Aγ,N}γ,N) =1

4

∥∥∥TrB[(∂γΓAγ,N )(ΓAγ,N )−1(∂γΓ

Aγ,N )]+TrB[(∂γΓAγ,N )(ΓAγ,N )−1(∂NΓAγ,N )]

TrB[(∂NΓAγ,N )(ΓAγ,N )−1(∂γΓAγ,N )] + 3 TrB[(∂NΓAγ,N )(ΓAγ,N )−1(∂NΓAγ,N )]

∥∥∥∞. (F4)

We have

∂γΓAγ,NRB =

−N 0 0 − 1

2√

1−γ0 N 0 00 0 1−N 0

− 12√

1−γ 0 0 − (1−N)

, and (F5)

∂NΓAγ,NRB = −γ (I2 ⊗ σZ) . (F6)

We then have

TrB[(∂γΓAγ,N )(ΓAγ,N )−1(∂γΓ

Aγ,N )] =

1N−γN + 1

1−N−4

4γ20

01

(γ−1)(N−1)+ 1N−4

4γ2

(F7)

TrB[(∂γΓAγ,N )(ΓAγ,N )−1(∂NΓAγ,N )] =

[− 1−2N

2γN(1−N)0

0 − 1−2N2γN(1−N)

](F8)

TrB[(∂NΓAγ,N )(ΓAγ,N )−1(∂γΓAγ,N )] =

[− 1−2N

2γN(1−N)0

0 − 1−2N2γN(1−N)

](F9)

TrB[(∂NΓAγ,N )(ΓAγ,N )−1(∂NΓAγ,N )] =

[1

N(1−N)0

0 1N(1−N)

]. (F10)

With this in place, it is straightforward to evaluate (F4).

2. SLD Fisher Information Value

The SLD Fisher information, unlike the RLD quantity, does not have a compact expression interms of the Choi operators of the channel family. Therefore, in this example, we make the choiceof a pure entangled probe state:

|ψp〉 =√p|00〉+

√1− p|11〉, (F11)

which gives us

ργ,N := Aγ,N(ψp) =

p(1− γN) 0 0

√p(1− p)(1− γ)

0 pγN 0 00 0 (1− p)γ(1−N) 0√

p(1− p)(1− γ) 0 0 (1− p)(1− (1−N)γ)

. (F12)

21

The SLD Fisher information takes the form of a 2× 2 matrix:

IF (γ,N ; {ργ,N}γ,N)jk = Tr [ργ,NLjLk] (F13)

where j and k each take values 1 or 2 that correspond to either γ or N . We have Lj defined implicitlyvia

∂jργ,N =1

2(ργ,NLj + Ljργ,N) . (F14)

Consider that ργ,N has spectral decomposition

ργ,N =∑j

λjγ,N |ψjγ,N〉〈ψ

jγ,N |. (F15)

We have

Lγ = 2∑

j,k:λjγ,N+λkγ,N>0

〈ψjγ,N |(∂γργ,N)|ψkγ,N〉λjγ,N + λkγ,N

|ψjγ,N〉〈ψkγ,N | and (F16)

LN = 2∑

j,k:λjγ,N+λkγ,N>0

〈ψjγ,N |(∂Nργ,N)|ψkγ,N〉λjγ,N + λkγ,N

|ψjγ,N〉〈ψkγ,N | (F17)

which enables us to calculate the elements of the SLD Fisher information matrix given in (F13).The expression that we evaluate is the right-hand side of the following Cramer–Rao bound:

Tr[WCov(θ)] ≥ Tr[WIF (θ; {ρθ}θ)−1] (F18)

where θ is a vector of unknown parameters of length D, and W is a D × D positive semidefiniteweight matrix. This is the Helstrom CRB, which is known to be attainable up to a constant factor[38, 40]. Again, we choose W = 1

4

(1 11 3

)just as we did in the case of calculating the RLD Fisher

information. In more detail, we evaluate

Tr[WCov({γ,N})] ≥ minp∈[0,1]

Tr[W [IF ({γ,N}; {Aγ,N(ψp)}{γ,N})]−1]. (F19)

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